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  <div class="section" id="numpy-polynomial-hermite-e-hermeint">
<h1>numpy.polynomial.hermite_e.hermeint<a class="headerlink" href="#numpy-polynomial-hermite-e-hermeint" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.polynomial.hermite_e.hermeint">
<code class="sig-prename descclassname">numpy.polynomial.hermite_e.</code><code class="sig-name descname">hermeint</code><span class="sig-paren">(</span><em class="sig-param">c</em>, <em class="sig-param">m=1</em>, <em class="sig-param">k=[]</em>, <em class="sig-param">lbnd=0</em>, <em class="sig-param">scl=1</em>, <em class="sig-param">axis=0</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/polynomial/hermite_e.py#L653-L772"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.polynomial.hermite_e.hermeint" title="Permalink to this definition">¶</a></dt>
<dd><p>Integrate a Hermite_e series.</p>
<p>Returns the Hermite_e series coefficients <em class="xref py py-obj">c</em> integrated <em class="xref py py-obj">m</em> times from
<em class="xref py py-obj">lbnd</em> along <em class="xref py py-obj">axis</em>. At each iteration the resulting series is
<strong>multiplied</strong> by <em class="xref py py-obj">scl</em> and an integration constant, <em class="xref py py-obj">k</em>, is added.
The scaling factor is for use in a linear change of variable.  (“Buyer
beware”: note that, depending on what one is doing, one may want <em class="xref py py-obj">scl</em>
to be the reciprocal of what one might expect; for more information,
see the Notes section below.)  The argument <em class="xref py py-obj">c</em> is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series <code class="docutils literal notranslate"><span class="pre">H_0</span> <span class="pre">+</span> <span class="pre">2*H_1</span> <span class="pre">+</span> <span class="pre">3*H_2</span></code> while [[1,2],[1,2]]
represents <code class="docutils literal notranslate"><span class="pre">1*H_0(x)*H_0(y)</span> <span class="pre">+</span> <span class="pre">1*H_1(x)*H_0(y)</span> <span class="pre">+</span> <span class="pre">2*H_0(x)*H_1(y)</span> <span class="pre">+</span>
<span class="pre">2*H_1(x)*H_1(y)</span></code> if axis=0 is <code class="docutils literal notranslate"><span class="pre">x</span></code> and axis=1 is <code class="docutils literal notranslate"><span class="pre">y</span></code>.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>c</strong><span class="classifier">array_like</span></dt><dd><p>Array of Hermite_e series coefficients. If c is multidimensional
the different axis correspond to different variables with the
degree in each axis given by the corresponding index.</p>
</dd>
<dt><strong>m</strong><span class="classifier">int, optional</span></dt><dd><p>Order of integration, must be positive. (Default: 1)</p>
</dd>
<dt><strong>k</strong><span class="classifier">{[], list, scalar}, optional</span></dt><dd><p>Integration constant(s).  The value of the first integral at
<code class="docutils literal notranslate"><span class="pre">lbnd</span></code> is the first value in the list, the value of the second
integral at <code class="docutils literal notranslate"><span class="pre">lbnd</span></code> is the second value, etc.  If <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">==</span> <span class="pre">[]</span></code> (the
default), all constants are set to zero.  If <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">==</span> <span class="pre">1</span></code>, a single
scalar can be given instead of a list.</p>
</dd>
<dt><strong>lbnd</strong><span class="classifier">scalar, optional</span></dt><dd><p>The lower bound of the integral. (Default: 0)</p>
</dd>
<dt><strong>scl</strong><span class="classifier">scalar, optional</span></dt><dd><p>Following each integration the result is <em>multiplied</em> by <em class="xref py py-obj">scl</em>
before the integration constant is added. (Default: 1)</p>
</dd>
<dt><strong>axis</strong><span class="classifier">int, optional</span></dt><dd><p>Axis over which the integral is taken. (Default: 0).</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.7.0.</span></p>
</div>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>S</strong><span class="classifier">ndarray</span></dt><dd><p>Hermite_e series coefficients of the integral.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>ValueError</strong></dt><dd><p>If <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">&lt;</span> <span class="pre">0</span></code>, <code class="docutils literal notranslate"><span class="pre">len(k)</span> <span class="pre">&gt;</span> <span class="pre">m</span></code>, <code class="docutils literal notranslate"><span class="pre">np.ndim(lbnd)</span> <span class="pre">!=</span> <span class="pre">0</span></code>, or
<code class="docutils literal notranslate"><span class="pre">np.ndim(scl)</span> <span class="pre">!=</span> <span class="pre">0</span></code>.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="numpy.polynomial.hermite_e.hermeder.html#numpy.polynomial.hermite_e.hermeder" title="numpy.polynomial.hermite_e.hermeder"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hermeder</span></code></a></p>
</div>
<p class="rubric">Notes</p>
<p>Note that the result of each integration is <em>multiplied</em> by <em class="xref py py-obj">scl</em>.
Why is this important to note?  Say one is making a linear change of
variable <img class="math" src="../../_images/math/f1149e7c6f7de3e7e6f56038c4c47b6b467bac2a.svg" alt="u = ax + b"/> in an integral relative to <em class="xref py py-obj">x</em>.  Then
<img class="math" src="../../_images/math/be769e642dd1ec3e3072c5506e4d73ded7b4b018.svg" alt="dx = du/a"/>, so one will need to set <em class="xref py py-obj">scl</em> equal to
<img class="math" src="../../_images/math/45ed8db4c9010e4d33a861dad6a7ef54c30608da.svg" alt="1/a"/> - perhaps not what one would have first thought.</p>
<p>Also note that, in general, the result of integrating a C-series needs
to be “reprojected” onto the C-series basis set.  Thus, typically,
the result of this function is “unintuitive,” albeit correct; see
Examples section below.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">numpy.polynomial.hermite_e</span> <span class="kn">import</span> <span class="n">hermeint</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">hermeint</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span> <span class="c1"># integrate once, value 0 at 0.</span>
<span class="go">array([1., 1., 1., 1.])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">hermeint</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">m</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># integrate twice, value &amp; deriv 0 at 0</span>
<span class="go">array([-0.25      ,  1.        ,  0.5       ,  0.33333333,  0.25      ]) # may vary</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">hermeint</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">k</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># integrate once, value 1 at 0.</span>
<span class="go">array([2., 1., 1., 1.])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">hermeint</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">lbnd</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># integrate once, value 0 at -1</span>
<span class="go">array([-1.,  1.,  1.,  1.])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">hermeint</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">m</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">lbnd</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">array([ 1.83333333,  0.        ,  0.5       ,  0.33333333,  0.25      ]) # may vary</span>
</pre></div>
</div>
</dd></dl>

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